Tagged Questions

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Equivalence of Axiom of Regularity

So Axiom of regularity states: every non-empty set A contains an element that is disjoint from A I'm wondering if this is equivalent as any set is not a member of itself? If so, how do we prove it? ...
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How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
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What is the weakest notion of “set” that we need, so that we can say the Yoneda lemma implies something about sets?

We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central ...
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Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
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Deriving the Axiom of Infinity

I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ...
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What is meant by one axiom being “weaker” than another?

If we have two axioms $A$ and $B$, what exactly is meant by axiom $A$ being weaker than axiom $B$? This question is a follow-up to A weaker Axiom of Infinity?
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A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at ...
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Replacing Axiom of Extensionality with a logical formalism

Is it possible to replace the Axiom of Extensionality with a formalism from logic, namely the following one: $\forall a \forall b (a=b\Leftrightarrow \forall P (P (a)\Leftrightarrow P (b)))$ ($P$ is ...
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If we abandon the axiom of regularity, can the cumulative hierarchy just become a definition?

In ZFC, the axiom of regularity is used to prove that every set is an element of some stage of the cumulative hierarchy. The index of the least such stage is, by definition, the rank of that set. Now ...
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Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
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What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
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Which axioms of ZFC or PA are known to not be derivable from the others?

Which, if any, axioms of ZFC are known to not be derivable from the other axioms? Which, if any, axioms of PA are known to not be derivable from the other axioms?
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An Exercise in Kunen (A Model for Foundation, Pairing,…)

This is exercise I.4.18 in Kunen's Set Theory. Derive $\forall y (y \notin y)$ from the Axioms of Comprehension and Foundation. Don't use the Pairing or Extensionality Axioms. Then find a 2 element ...
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What's the differences between naive and axiomatic set theory?

I'm at a loss studying math.Recently I decided to begin with set theory as it seems the most fundamental for math.I found the book Naive Set Theory by Halmos,and began to read it because it's so thin ...
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Understanding the Definition of the Axiom Schema of Specification

Consider the Axiom Schema of Separation: If $P$ is a property (with paramter $p$), then for any $X$ and $p$ there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those $u \in X$ that ...